Sphere bundle

Summary

A sphere bundle is a fiber bundle whose fibers are spheres of a fixed dimension. The main source of examples is the unit sphere bundle of a real vector bundle with an inner product. Sphere bundles carry the Euler class and the Gysin sequence, and the Hopf fibration is a circle bundle over the $2$-sphere. Milnor's exotic $7$-spheres were found as total spaces of $S^3$-bundles over $S^4$.